Derived as part of the answer to part (a), used in part (b).
The whole question is about you ability to turn words into equations and vise versa.
Consider: what does "rate of population growth" mean?

Here is your question:
Q8. A territory will support a maximum population of ##P_0##. Let the ratio of the population ##P## to the maximum population be ##p##... the rate of change of this ratio is the product of ##p## and the difference between ##p## and ##1##.

(a) write down the differential equation in ##p##
(b) show that the growth of the population is greatest when ##P=\frac{1}{2}P_0##
(c) the population starts at ##\frac{1}{4}P_0## and reaches ##\frac{1}{2}P_0## in 20 years.
Find the time for the population to reach ##\frac{7}{8}P_0##.
(d) find the value of ther maximum growth of the population to 3dp.

Derived as part of the answer to part (a), used in part (b).
The whole question is about you ability to turn words into equations and vise versa.
Consider: what does "rate of population growth" mean?

Here is your question:
Q8. A territory will support a maximum population of ##P_0##. Let the ratio of the population ##P## to the maximum population be ##p##... the rate of change of this ratio is proportional to the product of ##p## and the difference between ##p## and ##1##.

(a) write down the differential equation in ##p##
(b) show that the growth of the population is greatest when ##P=\frac{1}{2}P_0##
(c) the population starts at ##\frac{1}{4}P_0## and reaches ##\frac{1}{2}P_0## in 20 years.
Find the time for the population to reach ##\frac{7}{8}P_0##.
(d) find the value of ther maximum growth of the population to 3dp.

Does your text really say "defferential equation"? That makes me wonder about the grammar- and, in particular, whether "maximum growth" means the maximum population or maximum rate of growth of the population. If it is "maximum rate of growth", (b) asks you to "show that the growth of the population is greatest when [itex]P= \frac{1}{2}P_0[/itex]". What is the rate of growth then?